This is different than pigs and striders getting their saddle's data wiped, since horses, mules, and camels have menus.

Saddles that are modified (via Item Components) are reverted to non-modified saddles when dispensed onto a tamed horse.

How to reproduce:

Step 1: Give yourself a modified saddle using commands (or just rename a saddle.)
(Command is just an example)

Step 2: Put the modified saddle in a dispenser.

Step 3: Send a Redstone signal to the dispenser, dispensing the saddle onto the tamed horse

Step 4: Inspect the horse's inventory. The saddle has been reverted.

(This also happens to mules and camels)